To resolve it, a compressor \(\mathsf {Q}(\cdot ):\mathbb {R}^d\rightarrow \mathbb {R}^d\)  is unavoidable for each node to send compressed messages which can be usually encoded with a moderate number of bits. For example, the output of a binary quantizer (i.e. \(\mathsf {Q}(x)=1\)  if \(x\ge 0\)  and \(\mathsf {Q}(x)=-1\)  otherwise) needs only one bit to encode. Since \(\mathsf {Q}(\cdot )\)  is typically highly nonlinear and nonsmooth, how to design a provably convergent distributed algorithm with efficient communication compressions has attracted an increasing attention. To this end, quantized variants of the DGD [1]} have been proposed for convex problems [2]}, [3]}, [4]}, [5]}, [6]}, [7]} and non-convex problems [8]}, [9]}, [10]}, respectively. Unfortunately, their convergence rates are constrained by the sublinear rate of DGD. Though linear convergence has been achieved in [11]}, [12]}, [13]}, they are restricted to a specified compressor which is then extended to a class of stochastic compressors in [14]}, [15]}, [16]}. However, their stochastic compressors are assumed to be unbiased and \(\delta \) -contracted in the sense that \(\mathbb {E}[\mathsf {Q}(\mathbf {x})]=\mathbf {x}\)  and \(\mathbb {E}\Vert \mathsf {Q}(\mathbf {x})-\mathbf {x}\Vert ^2\le \delta \Vert \mathbf {x}\Vert ^2,\ \forall \mathbf {x}\in \mathbb {R}^d\) . Clearly, this condition excludes some important compressors, including deterministic compressors such as the binary quantizer. In fact, the unavoidable round-off error in finite-precision implementation can be regarded as the consequence of a biased compressor.
